0.1 Supremum & Infimum (Real Numbers)
The supremum and infimum are key concepts for defining completeness. In real analysis, we can use the [[order|ordering properties]] of \(\mathbb{R}\) to define the supremum and infimum of a set of real numbers.
The set of real numbers \(A\subset \mathbb{R}\) has an upper bound \(M \in \mathbb{R}\) such that \(M \leq M'\) for every upper bound \(M'\) of \(A\), then \(M\) is the least upper bound or supremum of \(A\), denoted \[ M=\sup{A}. \] Similarly, if \(A \subset \mathbb{R}\) has a lower bound \(m \in \mathbb{R}\) such that \(m\geq m'\) for every lower bound \(m'\) of \(A\), then \(m\) is the greatest lower bound or infimum of \(A\), denoted \[ m=\inf{A}. \]
If \(A=\{ x_{i}:i \in I \}\) is an indexed subset of \(\mathbb{R}\) we write \[ \sup{A}=\sup_{i \in I}{x_{i}},\quad \quad \inf{A}=\inf_{i \in I}{x_{i}}. \] Note that if \(M,m \in A\) then the supremum \(M\) and infimum \(m\) are equivalent to the maximum and minimum respectively. The key point is that the supremum and infimum need not belong to the set \(A\).
If a set \(A \subset \mathbb{R}\) is not bounded from above, then \(\sup_{}{A}= \infty\) and if \(A\) is not bounded from below then \(\inf_{}{A}=-\infty\).
0.2 Supremum & Infimum Properties
If \(A \subset \mathbb{R}\) then \(M = \sup_{}{A}\) if and only if: 1. \(M\) is an upper bound of \(A\); 2. \(\forall M'<M,~\exists x \in A~:~x>M'\).
Similarly, \(m = \inf_{}{A}\) if and only if: 1. \(m\) is a lower bound of \(A\); and 2. \(\forall m'>m,~\exists x \in A~:~x<m'\).
We frequently use this proposition as follows:
- If \(M\) is an upper bound of \(A\); then \(\sup_{}{A}\leq M\).
- If you \(A\) is nonempty and bounded from above, then for every \(\varepsilon>0,~\exists x \in A~s.t. x>\sup_{}{A}-\varepsilon\).